SOLUTION: According to Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots?
f(a)= a^5-4a^2-7
f(x) = 3x^3 + 9x^2
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Question 175797This question is from textbook
: According to Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots?
f(a)= a^5-4a^2-7
f(x) = 3x^3 + 9x^2 + 8x
This question is from textbook
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
maximum number of positive roots is number of times the sign of the coefficient changes for f(x).
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maximum number of negative roots is number of times the sign of the coefficient changes for f(-x).
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possible number of roots is:
number of maximum roots minus 2 until you get below 0.
example 1:
maximum number of roots is 6.
possible number of roots is 6,4,2,0
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example 2:
maximum number of roots is 7.
possible number of roots is 7,5,3,1
note that getting 0 roots in this case is not a possibility.
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if i let x = a, your first equation becomes:
f(x) = x^5 - 4x^2 - 7
i could have left the variable is a, but it graphs better when i call it x since some graphing calculators or online graphing calculators only recognize x as the variable.
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your first equation is:
f(x) = x^5 - 4x^2 - 7
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equation must be in standard form meaning that the degree of the exponents goes from higher to lower as you go from left to right.
this equation is already in that form as is your second equation so i won't bother to mention this again.
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signs of the coefficients in the first equation are:
"+ - -"
number of changes in the sign of the coefficients is 1.
maximum number of positive roots is 1.
possible number of positive roots is 1.
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to get negative roots, we need to get f(-x).
f(-x) = (-x)^5 - 4(-x)^2 - 7
this becomes:
f(-x) = -x^5 -4x^2 - 7
signs of the coefficients are:
"- - -"
maximum number of negative roots is 0.
possible number of negative roots is 0.
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graph of this equation is:
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your second equation is:
f(x) = 3x^3 + 9x^2 + 8x
it is in standard form.
signs of the coefficients are:
"+ + +"
number of sign changes is 0.
maximum number of positive roots is 0.
possible number of positive roots is 0.
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f(-x) = 3*(-x)^3 + 9*(-x)^2 + 8*(-x)
this is the same as:
-3x^3 + 9x^2 - 8x
signs of the coefficients are:
"- + -"
number of sign changes is 2.
maximum number of negative roots is 2.
possible number of negative roots is 2,0.
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graph of this equation is:
this particular equation has 0 negative roots and 0 positive roots.
it does have a root at x = 0.
this can be determined from the equation because x factors out and we are left with:
x * (3x^2 + 9x + 8) = f(x) = 0
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even though the maximum number of negative roots is 2, this equation had 0.
0 was one of the possible number of roots so descartes rule of signs is accurate.
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